ECEN4797 5797 Introduction to Power Electronics Another Compensator Design Example Power stage parameters iL t Iout L C Vg R vo Vref 1 8 V Iout 0 to 5 A Dead time control fs 1 MHz Switching frequency fs 1MHz PWM Vg 5 V duty cycle command L 1 H error Compensator Gc s Vref 1 VM RL 30 m C 200 F Point of Load Synchronous Buck Regulator Resr 0 8 m VM 1 V H 1 ECEN5797 Buck Averaged Small Signal Model Vg d RL L iL Gvd s 1 Resr vg D iL R D vg Io d 1 1 2 CL Qloss Gvd s Vg C Pair of poles fo v Q Qloss Qload s esr 1 s s Q o o Low frequency gain including PWM gain 11 kHz Gvdo L C 2 3 7 2 dB Resr RL v o d Qload R 5 L C Q Q loss load 2 3 7 2 dB Qloss Qload 1 5 14dB VM ESR zero 1 f esr 1 MHz 2 CResr ECEN5797 ECEE Department University of Colorado Boulder 2 ECEN4797 5797 Introduction to Power Electronics Uncompensated loop gain Tu iL t Iout L C Vg R Gvd s Dead time control fs 1 MHz vo Hsense 1 in this example duty cycle command error PWM Compensator 1 VM Gc s 1 Vref Tu s Hsense 1 VM Gvd s Plot magnitude and phase responses of Tu s to plan how to design Gc s ECEN5797 Magnitude and phase Bode plots of Tu 80dB Tu s Hsense 1 VM Gvd s 60dB 40dB 20dB Tuo Gvdo 1 VM H sense 5 14dB Q 2 3 7 2dB f o 11kHz 0dB 40dB dec 20dB 10 1 2Q f Tuo o fc fo 2 0o t target t fc 20dB dec f esr 1MHz 10 1 2Q f o 90o 1 10 f esr 180o 10 Hz 100 Hz 1 KHz 10 KHz 100 KHz 1 MHz ECEN5797 ECEE Department University of Colorado Boulder ECEN4797 5797 Introduction to Power Electronics Magnitude and phase Bode plots of Tu Exact magnitude and phase responses MATLAB Uncompensated loop gain Tu Gvd Hsense 1 VM magnitude db 50 0 Target cross over frequency fc fs 10 100 kHz 50 100 3 4 10 5 10 6 10 10 phase deg 0 100 No phase margin a lead PD compensator is required 200 3 4 10 5 10 frequency Hz 6 10 10 ECEN5797 Lead PD compensator design 1 Choose f c 100 kHz m 53o 2 Compute 33 kHz 300 kHz 3 Find Gco to position the crossover frequency 2 f Tuo o Gco fc fp fz 1 Gco 1 Tuo fc f o 2 fz 5 45 15 dB fp Magnitude Magnitude of Tu at fc of Gc at fc ECEN5797 ECEE Department University of Colorado Boulder ECEN4797 5797 Introduction to Power Electronics Lead PD compensator summary s 1 z 1 Gc s Gco 1 s 1 s p1 p 2 Lead compensator Gco 5 45 15 dB f z 33 kHz f p1 300 kHz f c 100 kHz 1 10 of fs HF pole High frequency gain of the lead compensator Gco fp1 fz 49 34 dB Added high frequency pole f p 2 1 MHz fesr fs in this example Practical implementation would require an op amp with a gain bandwidth product of at least 49 fp2 49 MHz ECEN5797 Loop gain with lead PD compensator 80dB s 1 z Gc s Gco 1 s p1 60dB 40dB Tuo Gco 28 7 29 7dB 1 1 s p2 20dB f c 100 kHz 0dB 20dB f z 10kHz f z 33kHz f p 300 kHz 0o 90o m 53o 180o 10 Hz ECEN5797 100 Hz 1 KHz 10 KHz 100 KHz 1 MHz ECEE Department University of Colorado Boulder ECEN4797 5797 Introduction to Power Electronics Add lag PI compensator Integrator at low frequencies Choose 10fL fc so that phase margin stays approximately the same fL 8 kHz Keep the same cross over frequency Gc Gco Gcm 5 45 15 dB ECEN5797 Adding PI Compensator 80dB 60dB 40dB 20dB f L 8kHz f c 100 kHz 0dB 20dB 0o 10 f L 1 10 f L 90o PI compensator phase m 53o 180o 10 Hz 100 Hz 1 KHz 10 KHz 100 KHz 1 MHz ECEN5797 ECEE Department University of Colorado Boulder ECEN4797 5797 Introduction to Power Electronics Complete analog PID compensator summary Gcm 5 45 15 dB f L 8 kHz f z 33 kHz f p1 300 kHz f p 2 1 MHz Crossover frequency f c 100 kHz 1 10 of fs o Phase margin m 53 ECEN5797 Magnitude and phase Bode plots of T 80dB 60dB 40dB 20dB f c 100 kHz 0dB 20dB Phase of uncompensated Tu 0o Phase of compensated T 90o m 53o 180o 10 Hz 100 Hz 1 KHz 10 KHz 100 KHz 1 MHz ECEN5797 ECEE Department University of Colorado Boulder ECEN4797 5797 Introduction to Power Electronics Verification exact loop gain magnitude and phase responses MATLAB Loop Gain magnitude db 50 f c 105 kHz 0 50 phase deg 100 0 3 4 5 6 50 m 551 6 o 100 150 200 250 3 4 10 5 10 frequency Hz 6 10 10 ECEN5797 Analog PID compensator implementation C3 C4 R4 R2 C2 output voltage sense R1 Vref vc control voltage Design equations approximate Gc s Gcm s 1 z 1 s 1 s p1 p 2 Gcm L 1 s fz fL R2 R1 1 2 R1 R4 C 4 f p2 1 2 R2C 2 f p1 1 2 R4 C 4 1 2 R2C3 ECEN5797 ECEE Department University of Colorado Boulder ECEN4797 5797 Introduction to Power Electronics Verification of closed loop responses Closed loop reference to output response Closed loop output impedance and step load transient response ECEN5797 Construction of closed loop T 1 T response 80dB Closed loop reference to output response v vref T 1 T 60dB 40dB 20dB v vref 0dB 20dB Closed loop BW fc 40dB 60dB 80dB 10 Hz 100 Hz 1 KHz 10 KHz 100 KHz 1 MHz ECEN5797 ECEE Department University of Colorado Boulder ECEN4797 5797 Introduction to Power Electronics Closed loop reference to output response Reference to output response 50 v vref T 1 T db 0 50 100 3 4 10 5 10 6 10 10 ECEN5797 Small signal step reference response 1 82 10 mV step 1 79 V to 1 8 V in vref 1 8 vo t 1 78 4 8 5 5 2 5 4 5 6 5 8 6 6 2 6 4 6 6 6 8 4 x 10 iL t 5 0 4 8 5 5 2 5 4 5 6 5 8 6 6 2 6 4 6 6 6 8 4 x 10 0 5 d t 0 4 8 5 5 2 20 s div 5 4 5 6 5 8 6 6 2 6 4 6 6 6 8 4 x 10 ECEN5797 ECEE Department University of Colorado Boulder ECEN4797 5797 Introduction to Power Electronics Small signal step reference response 1 82 10 mV step …